Abbas Khavaran

Vantage Partners, LLC, Cleveland, Ohio

Jet Surface Interaction—Scrubbing Noise

NASA/CR—2013-218090

September 2013
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Abbas Khavaran

Vantage Partners, LLC, Cleveland, Ohio

Jet Surface Interaction—Scrubbing Noise

NASA/CR—2013-218090

September 2013

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared under Contract NNC12BA01B
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Level of Review: This material has been technically reviewed by NASA technical management OR expert reviewer(s).

This report is a formal draft or working 

paper, intended to solicit comments and 

ideas from a technical peer group.

This report contains preliminary fi ndings, 

subject to revision as analysis proceeds.

Acknowledgments

This work was sponsored by the NASA Fundamental Aeronautics Program. The author is grateful to the Acoustics Branch at 

NASA Glenn Research Center for supporting this research effort, and to Dr. Stewart Leib for his very insightful comments and 

suggestions.

This work was sponsored by the Fundamental Aeronautics Program 

at the NASA Glenn Research Center.
Abstract 

Generation of sound due to scrubbing of a jet flow past a nearby solid surface is 

investigated within the framework of the generalized acoustic analogy theory. The analysis 

applies to the boundary layer noise generated at and near a wall, and excludes the scattered 

noise component that is produced at the leading or the trailing edge. While compressibility 

effects are relatively unimportant at very low Mach numbers, frictional heat generation and 

thermal gradient normal to the surface could play important roles in generation and 

propagation of sound in high speed jets of practical interest. A general expression is given 

for the spectral density of the far-field sound as governed by the variable density PridmoreBrown equation. The propagation Green’s function should be solved numerically starting 

with the boundary conditions on the surface and subject to specified mean velocity and 

temperature profiles between the surface and the observer. The equivalent sources of 

aerodynamic sound are associated with non-linear momentum flux and enthalpy flux terms 

that appear in the linearized Navier-Stokes equations. These multi-pole sources should be 

modeled and evaluated with input from a Reynolds-Averaged Navier-Stokes (RANS) 

solver with an appropriate turbulence model.

1.0 Introduction 

The Fundamental Aeronautics program at NASA faces new technological challenges in order to 

meet recent air transportation rules that include reduced noise, emission, and fuel consumption. 

The U.S. commercial aviation is projected to increase by 70% in the period 2010 – 2030, 

resulting in more exposure to air traffic noise in the communities surrounding airports. Recent 

noise metrics require a cumulative (cutback, sideline, and approach) noise reduction in the 

subsonic transport of 32dB by 2015 and 52dB by 2025 (referenced to 737-800 aircraft with 

CFM56-7B engines). In 2008, NASA instigated a call for conceptual design of future 

commercial transport in order to meet specific goals related to noise, air pollution, and fuel 

consumption. Early studies indicated that both propulsion system and engine placement have to 

be modified in order to achieve these new objectives. In particular, placement of the aircraft 

engine for the purpose of reduced community noise poses new challenges in structural design, 

material selection as well as computations. Configurations such as over the wing engine mount 

(Fig. 1), distributed propulsion, Hybrid Wing Body (HWB) concept, and/or high aspect-ratio 

rectangular exhaust geometry with extended beveled surfaces designed to shield noise from 

reaching the ground, also produce new noise sources due to scrubbing of the flow past structure, 

skin vibration, boundary layer noise, and edge noise due to the scattering of sound from sharp 

edges. 

In 2010, a series of tests were initiated at NASA Glenn Research Center in order to study the 

propulsion/airframe integration. These experiements, under the umbrella of “Jet-Surface 

NASA/CR—2013-218090 1

Jet Surface Interaction—Scrubbing Noise 

Abbas Khavaran 

Vantage Partners, LLC 

Cleveland, Ohio 44135 
Interaction Test (JSIT),” were intended to guide analytical studies towards prediction model 

development. The initial tests consisted of placement of a flat plate in the proximity of a round 

jet. Details of the experimental setup and test cofigurations are provided by C. Brown [Ref. 1], 

and G. Podboy [Ref. 2]. Measurements were gathered on both sides of the plate, commonly 

referred to as the reflected and shielded sides. In addition to the far-field microphone noise 

measurements [1], phased array source localization experiments [2] were carried out in order to 

shed light on the noise source location and its frequency content under different exhaust 

conditions. 

An example of JSIT noise data that shows 

power specral density per Strouhal frequency 

in a 2-in diameter (D =5.08cm) Mach 0.90 

unheated jet, as reported by Brown in Ref. 1, 

is reproduced in figure 2. Here a solid surface 

extending 12-diameters from jet exit in the 

downstream direction was positioned at D/2 

from the jet centerline. Far-field measurements 

are shown at 90o and at 100D for a jet in 

isolation (no solid surface), as well as shielded 

and reflected sides of the jet in the presence of 

the surface. 

An extensive database has been generated as a function of the jet operating conditions and the 

length and position of the nearby solid surface. Similarly, an example of source localization data 

for the above jet is reported by Podboy as figure 17 of Ref. 2 (not shown here). It presents a 

beam-form map of the source location for an isolated jet as well as when surfaces of varying 

length are placed between jet and the phased array.

It is commonly accepted that the 

trailing edge noise is responsible for 

the enhanced low-frequency and 

peak noise amplitude measured on 

both reflected and shielded sides of 

the plate as seen in figure 2. A 

distinctive feature of the trailing 

edge noise component is its cardioid 

directivity shape and a dependence 

on jet velocity to the fifth power 

[Refs. 3, 4]. Recent measurements 

by Lawrence et al. [Ref. 5] indicate 

that the velocity power factor could 

be as large as 6 at longer plate 

length L that yield a wetted trailing 

edge. FFowcs Williams and Hall 

[Ref. 3] describe a solution to the 

edge noise from a rigid plane in the 

presence of a turbulent flow using 

Lighthill’s acoustic analogy. They 

 Figure 1. Jet surface interaction.

 Figure 2. Sample JSIT noise measurement as

 reported by C. Brown in Ref. 1.

NASA/CR—2013-218090 2
consider a very low Mach number flow, and take advantage of an appropriate Green’s function 

applicable to a point source in the vicinity of a half-plane [Ref. 6]. Howe [Ref. 7] reviews 

various theoretical approaches in understanding the trailing-edge noise, and summarizes diverse 

theories into a single model that exhibits the dependence of the acoustic field on the turbulent 

fluctuations near the edge as well as the significance of the Kutta condition at this point. 

Goldstein [Ref. 8] includes the effect of the mean velocity gradient on the trailing edge noise by 

imposing a convecting vortex source, or gust, of the general form on an 

incompressible transversely sheared mean flow.

Our interest here is the turbulence-generated noise due to the scrubbing of the jet flow past an 

adjacent flat surface. At very low Mach number, Howe [Ref. 9] describes the far field acoustic 

domain in terms of the wall-pressure wave number-frequency spectrum. Various models have 

been suggested for wall-pressure spectrum [Ref. 10] that are used in conjunction with a 2D or 3D 

free-space Green’s function to express the far-field spectral density. These approximations 

ignore the mean flow refraction, compressibility effects, and sound produced outside the 

Boundary Layer (BL) at distances that are not small relative to the acoustic wavelength. At jet 

velocities of practical interest typical of a jet exhaust, frictional heat generation and thermal 

gradients in the BL could play important roles in generation as well as propagation of sound. 

Additionally, skin vibration could influence the surface boundary conditions and the subsequent 

radiated sound. 

In section 2 we derive an expression for the propagation Green’s function (GF) applicable to the 

scrubbing noise on a flat surface. The GF needs to be obtained numerically when the mean flow 

profiles are non-uniform. An expression for the far-field pressure is given in section 4. Ideally a 

RANS solution for the flow in the proximity of the surface provides the mean flow and 

turbulence information required in evaluating both the source strength and the GF. Sample GF 

computations using analytical profiles for the mean flow and a summary are provided in sections 

6 and 7. 

2.0 Formulation of the Scrubbing Noise 

 Consider turbulence-generated noise due to the scrubbing of a jet flow with an adjacent flat 

plane (Fig. 3). The mean flow is represented as a unidirectional transversely sheared flow 

(independent of x1 and x2 directions). We consider the mean static pressure p within the BL to 

be a constant. At very low Mach numbers the compressibility effects may be negligible, 

however at Mach numbers typical of a jet exhaust, frictional heat generation and thermal 

gradients in the BL could play important roles in generation as well as propagation of sound. 

The RANS solution provides such effects via the action of viscosity near the surface. Hence the 

noise generation process due to the scrubbing of a jet with a nearby surface is, in general, 

described by full acoustic analogy as considered in Appendix-A, and by Pridmore-Brown 

equation within a locally parallel flow approximation [Ref. 11] 

Lπ′ = Γ, π′ ≡ p′(



x,t)

γ p

 (1) 

where is the operator 

Ω(x1 /U(x3 )− t, x2 , x3 )

L

NASA/CR—2013-218090 3
L ≡ D D2 − ∂

∂xj

(c

2 ∂

∂xj

) ⎛

⎝

⎜

⎞

⎠

⎟ + 2c

2 ∂U

∂xj

∂2

∂x1 ∂xj

, D ≡ ∂

∂t

+U ∂

∂x1

 , (2) 

and the source term on the right hand side of (1) is defined according to the generalized 

acoustic analogy [Ref. 12] 

Γ = 1

γ p −D ∂

∂xi

(c

2 ∂eij

∂xj

)+ 2c

2 ∂U

∂xj

∂2eij

∂x1 ∂xi

+ D2Q ⎛

⎝

⎜

⎞

⎠

⎟ , (3a) 

Q = −(γ −1) 1

2

D(ρv′

2 )+

∂

∂xj

(ρv′

j

h′

o )+ (ρv′1v′j)

∂U

∂xj

⎛

⎝

⎜

⎞

⎠

⎟ , (3b) 

stress tensor eij is the difference between the fluctuating and Favre-averaged Reynolds stresses 

eij = −(ρv′

i

v′

j − ρv′i

v′

j), (3c) 

and h′

o denotes the fluctuations in the moving-frame stagnation enthalpy (see the nomenclature). 

 Figure 3. Jet scrubbing on a nearby surface.

The mean velocity U and density ρ are functions of the normal coordinate (x3 >0), and c is the 

mass-averaged speed of sound 

U = U(x3 ), c

2 = c2

(x3 ) = γ p / ρ(x3 ). (4) 

Note that, in practice, both the mean flow and turbulence depend on the transverse direction x2 as 

the source strength eventually diminishes at some distance ±L2 away from the jet centerline. In 

the stream-wise direction, the scrubbing effect may be limited to a distance 0 ≤ x1 < L1 , or the jet 

may persist beyond the plate edge at L1 where interaction of the turbulence with the trailing edge 

produces edge noise. Here, in formulating the scrubbing noise problem, we limit the source to a 

Γ(



x,t)

NASA/CR—2013-218090 4
finite x2 span by considering the mean flow as locally parallel in both x1 and x2 directions – thus 

express the problem in three dimensions. This assumption, although more suited to jets with a 

high aspect-ratio rectangular exhaust, permits source-volume integration within the wetted span 

in x2 direction. A round jet may be transformed into a rectangular strip by using an appropriate 

conformal mapping transformation. When all changes in span-wise x2 direction are neglected, the 

sound intensity should be expressed as per unit thickness. The special 2D case will be discussed 

in section 5. 

We seek a solution to equation (1) of the form 

π′(



x,t) = G(



x,t;



y,τ )Γ(



y,τ

τ

∫ 

y

∫ )dτd



y , (5) 

where G denotes the GF 

LG(



x,t;



y,τ ) = δ (



x − y)δ (t −τ ) . (6) 

In a bounded media, the above form of the solution requires for the GF to satisfy certain 

boundary conditions on the plate. The source-volume integration in (5) is carried out over noisegenerating region 

y in the vicinity of the scrubbing surface and at the emission time τ . The 

solution described in the following excludes the trailing edge noise that travels to the opposite 

side of the surface as alluded to earlier. This latter noise component requires a Wiener-Hopf 

type solution. 

We define a Fourier Transform (FT) pair with respect to (x1, x2 ) coordinates as well as time t. 

The corresponding wave number vector is denoted as 

kt = (k1, k2 ) 

 

Gˆ(



kt , x3, y3,ω) = 1

(2π )

3 G(



x,t;



y,τ )

−∞

∞

∫ −∞

∞

∫ −∞

∞

∫ e−i[ k1 (x1−y1)+k2 (x2−y2)−ω (t−τ )]

d(x1−y1 )d(x2−y2 )d(t−τ )

G(



x,t;



y,τ ) = Gˆ(



kt , x3, y3,ω)

−∞

∞

∫ −∞

∞

∫ −∞

∞

∫ e

i[ k1 (x1−y1)+k2 (x2−y2)−ω (t−τ )]

dk1dk2dω .

 (7) 

Throughout, an over hat carrot is used to denote a FT variable as defined above. 

The transform of equation (6) is

∂2

Gˆ

∂x3

2 + (c

2

)′

c

2 − 2k1

U′

−ω + k1

U

⎛

⎝

⎜

⎞

⎠

⎟

∂Gˆ

∂x3

+ (−ω + k1

U)

2

c

2 − k1

2 − k22 ⎛

⎝

⎜

⎞

⎠

⎟ Gˆ = i

(2π )

3

δ (x3− y3 )

c

2

(−ω + k1U)

 , (8) 

where prime as used on mean variables c

2 and U denotes differentiation with respect to x3. The 

equation possesses a critical layer at x3 where factor −ω + k1

U(x3 ) vanishes. To avoid this

NASA/CR—2013-218090 5
singularity we require k1 <ω /Ue where Ue denotes some peak mean velocity in the BL. This 

upper limit for k1 is usually referred to as the non-convective domain for the wave number. As 

discussed in [Refs. 13, 14], in the convective domain k1 >ω /Ue it becomes necessary to provide 

a loss mechanism, such as turbulent viscosity, in order to evade the singularity. The dominant 

value of k1 contributing to the far-field noise will be determined in section 3. It will be shown 

that when Ue < c∞ the upper limit condition stated for k1 will be satisfied at all observer location. 

For convenience, the second-order linear differential equation (8) is rearranged as

∂2 

G

∂x3

2 + f (



kt , x3,ω)



G = δ (x3− y3 ) . (9) 

Since equation (9) is self-adjoint, the newly defined GF, , is symmetric in (x3, y3), i.e., 

G(



kt , x3, y3,ω) = 

G(



kt , y3, x3,ω) and is related to the GF of interest as

Gˆ(



kt , x3, y3,ω) = i

(2π )

3

1

c(y3 )c(x3 )

−ω + k1U(x3 )

−ω + k1

U(y3 ( ))

2



G(



kt , x3, y3,ω) , (10) 

where 

f (



kt , x3,ω) = χ2 − 2 − 2(k1

U′)

2

(−ω + k1U)

2 +

k1

U′′

−ω + k1

U

+ (c

2

)′

c

2

k1

U′

−ω + k1

U ,

χ2(



kt , x3,ω) = (−ω + k1

U)

2

c

2 − k1

2 − k22 ,

2

(x3 ) = 1

2

∂

∂x3

(c

2

)′

c

2

⎛

⎝

⎜

⎞

⎠

⎟ +

1

4

(c

2

)′

c

2

⎛

⎝

⎜

⎞

⎠

⎟

2

.

 (11) 

and (c

2

)′ / c

2 = (∂ρ / ∂x3 )/ ρ . 

It should be noted that unlike the present problem, the propagation equation might not, in 

general, reduce to a self-adjoint from. Such is the case when the governing equations are written 

for a spreading flow [Ref. 15]. In that case, working with the adjoint GF will entail additional 

integrals that should be carried out on the surfaces. A general solution for the five field variables 

using a vector GF formulation is provided in Appendix A. The second integral in equation 

(A16) in the appendix describes the contribution to the acoustic field due to the nearby surfaces. 

In the formulation described herein, the GF satisfies conditions that make the surface integral 

equal to zero. 

Far away from the noise generating region, as x3 → ∞ the mean velocity and the sound speed 

approach their respective ambient values U∞ and c∞ , and equations (9) and (11) show that 

 

∂2 

G

∂2

x3

2 + χ ∞

2 

G = 0, x3 → ∞ (12a) 



G

Gˆ

NASA/CR—2013-218090 6
where 

χ ∞

2 = (−κ o + k1M∞ )

2 − k1

2 − k22 ; κ o = ω / c∞ , M∞ = U∞ / c∞ (12b) 

The Mach number M∞ takes account of the flight effect when surface is in motion relative to the 

ambient. The far-field solution to equation (12a) is 



G(



kt , x3, y3,ω) = b e−iχ ∞x3 , x3 → ∞ (13a) 

where complex number b has dimension of length, and in general depends on ω, k1, k2 . An 

outgoing propagating wave requires the negative root of χ ∞

2 when χ ∞2 > 0 , and when χ ∞2 is 

negative the branch-cut is chosen such that χ ∞ = −i(| χ ∞

2 |)

1/2 . When χ ∞

2 < 0 the pressure decays 

exponentially normal to the surface (evanescent waves). Aerodynamic noise scattered off the 

edge of a half-plane is sometimes described [Ref. 16, 17] as the interaction of evanescent waves 

(rather than propagating acoustic waves) with the edge. These disturbances are dominated by 

spectral components that possess a subsonic phase velocity in a constant x3 plane. 

Since we are interested in the scrubbing noise that reaches a distant observer, the radiation field 

would be dominated by wave numbers, in constant x3 plane, that possess a supersonic phase 

velocity 

ω

M∞k1 + k1

2 + k22

> c∞ (13b) 

 

The far-field boundary condition for a bounded solution to equation (9) requires 

∂



G

∂x3

+ iχ ∞



G = 0, x3 → ∞ , (14) 

and a no slip boundary condition on the scrubbing surface (U = 0, x3 → 0) combined with the x3

component of the linearized momentum equation within the turbulent boundary layer (TBL) 

shows that 

∂(ρv′

3 )

∂t

+

∂p′

∂x3

= 0, x3 → 0 (15) 

Upon replacing ρ with the mean fluid density ρo near the surface, we apply a FT to the latter 

equation to obtain 

−iωρo v′

3

 +

∂p′

∂x3

= 0, x3 → 0 (16) 

Assuming that Fourier components of the normal velocity and pressure on the surface relate 

through the surface impedance Z(ω) as

NASA/CR—2013-218090 7
p′ = Z v′

3

 , x3 = 0 (17) 

we find 

∂π

′

∂x3

− iρoω

Z

π

′ = 0, x3 = 0 . (18) 

Since the FT of a convolution is the product of the Fourier transforms, equations (5) implies 

π

′(



kt , x3,ω) = (2π )

3 Gˆ(



kt , x3, y3,ω)Γˆ(



kt , y3,ω)dy3

y3

∫ , (19) 

and upon using (19) in (18) we find 

∂Gˆ

∂x3

− iρoω

Z

Gˆ = 0, x3 = 0 (20) 

Our interest is in the boundary condition applicable to 

G rather than Gˆ . Substituting equation 

(10) in (20) shows that 

∂



G

∂x3

−ψ 

G = 0, x3 = 0 (21a) 

ψ (k1,ω,Z) = iκ o

Z

c∞

2

co

2 +

c′

o

co

+

k1

ω

U′(0) ⎛

⎝

⎜

⎞

⎠

⎟ , Z(ω) ≡ Z(ω)

ρ∞c∞

 (21b) 

where co is the mean sound speed in the fluid at the surface interface, and c′

o is its derivative at 

x3 = 0. An alternative way of deriving the surface boundary condition (21a) is described in 

Appendix B. 

The problem at hand is thus reduced to solving equation (9) subject to the two boundary 

conditions (14) and (21). 

The two boundary conditions could be expressed in a more general form as 

αi

∂π

′

∂x3

+ βiπ

′ = γ i , i = 1,2 (22) 

where i=1 and 2 refer to the boundary points at x3 = 0 and ∞. A non-zero value of γ i (such as 

presence of a source term in the linearized momentum equation near the surface) would 

contribute to the homogeneous solution to the differential equation. However, for our problem 

we consider γ 1 = γ 2 = 0 , thus the homogeneous solution is identically zero, and we are left with 

the particular solution only. 

NASA/CR—2013-218090 8
The GF in equation (9) may be expressed in terms of two linearly independent solutions 

Vi(



kt , x3,ω) to the homogeneous equation 

∂2Vi / ∂x3

2 + f Vi = 0, i = 1,2 (23a) 

such that V1 and V2 satisfy the homogeneous boundary conditions at 0 and ∞, respectively. For 

brevity we omit 

kt and ω from the arguments of V1 and V2. 

∂V1(x3 )

∂x3

−ψV1(x3 )= 0, x3 = 0 (23b) 

∂V2 (x3 )

∂x3

+ iχ ∞V2 = 0, x3 → ∞ (23c) 

and 



G(



kt , x3, y3,ω) = V2 (x3 )V1(y3 )/W (y3 ), y3 < x3

V1(x3 )V2 (y3 )/W (y3 ), y3 > x3

(24) 

andW V1,V2 (y3 ( )) is the Wronskian. For convenience we normalize both V1 and V2 at zero 

argument y3 = 0 such that 

V1(0) = V2 (0) = 1, (25) 

and since according to Abel’s formula the Wronskian to equation (23a) is independent of y3 , 

then we evaluateW = Wo at the surface. Equations (23b,c) and (25) conclude that 

Wo (



kt ,ω,Z) = V′

2 (0)−ψ (k1,ω,Z). (26) 

The above expression shows that the Wronskian Wo depends on the BL velocity and temperature 

profiles at y3 = 0 , as well as the surface impedance Z through parameter ψ . It is noted that 

equation (23a) supports Helmholtz instabilities that would arise when 

kt and ω satisfy the 

Eigen-wave condition Wo (



kt ,ω,Z) = 0 . The instability waves could be triggered when the BL 

profile supports the Eigen-wave condition. These waves could grow exponentially and dominate 

the region they occupy. Subsequently we need to set the wave number limit to avoid these 

special cases. For example, when the surface is rigid and the BL is considered as incompressible 

such that ψ (k1,ω,Z) = k1

U′(0)/ω , we should requireV′

2 (0)/U′(0) >1/Ue for the Wronskian Wo

to remain non-zero for all wave numbers within the non-convective range k1 <ω /Ue . However 

as we will discuss shortly,V′

2 (0) is not known a priori and is determined only after equation (20a) 

is solved as a boundary value problem. Jones [Ref. 18] studied the instability of a twodimensional shear layer using a model velocity profile in which the velocity increased linearly 

from 0 to Ue over a distance of h from the surface and then remained constant. He showed that 

NASA/CR—2013-218090 9
the onset of Helmholtz instabilities occur at a Strouhal frequency of ωh /Ue 1/ 3 where ω

denotes the frequency of sound waves falling on the shear layer. At frequencies above this 

critical value the layer remained stable. 

Upon placing (24) into (10), the Fourier transformed GF at an observer point above the source 

region is given as

Gˆ(



kt , x3, y3,ω) = i

(2π )

3

1

c(y3 )c(x3 )

−ω + k1U(x3 )

−ω + k1

U(y3 ( ))

2

V2 (x3 )V1(y3 )

Wo

, x3 > y3 (27) 

An alternative form of the GF valid for x3< y3 (see equation 24) should be used if one were 

interested in the surface pressure. The two functions Vi are determined numerically. The first 

variable is evaluated when equation (23a) is solved as an initial value problem withV1(0) = 1 and 

V′

1 (0) =ψ . The second variable is determined when (23a) is solved as a boundary value problem 

subject to V2 (0) = 1, and V′

2 (x3 )+ iχV2 (x3 ) = 0 as x3 → ∞ . Fortunately, as we shall see next, 

there are only certain values of wave numbers k1 and k2 that make the major contribution to the 

radiated far-field noise. 

3.0 Stationary Phase Method 

 We apply an inverse FT with respect to wave numbers k1 and k2 and write the GF as 

G(



x,



y;ω) = Gˆ(



kt , x3, y3,ω)

−∞

∞

∫ −∞

∞

∫ e

i[ k1 (x1−y1)+k2 (x2−y2)]

dk1dk2 . (28) 

Since the interest is in the far-field noise, the appropriate from for V2 is

V2 (



kt , x3,ω) = b2e−iχ ∞x3 , x3 → ∞ . (29) 

Complex number b2 (



kt ,ω) has a dimension of unity, and should be a known parameter once the 

boundary value problem (23a) is solved for V2. Placing (27) and (29) into (28), the double 

integral involving wave numbers k1 and k2 becomes 

G(



x,



y;ω) = i

(2π )

3

1

c(y3 )c(x3 )

−ω + k1U(x3 )

−ω + k1

U(y3 ( ))

2

b2V1(



kt , y3,ω)

Wo (



kt ,ω,Z) k2

∫

k1

∫ e

iΘ(



kt ,



x,ω )

dk1dk2 , (30) 

where 

Θ(



kt ,



x,ω) = k1(x1 − y1 )+ k2 (x2 − y2 )− χ ∞ x3 . (31) 

We adopt a spherical coordinate system where θ is the angle the observer position vector makes 

with the direction of the mean flow (Fig. 4) 

NASA/CR—2013-218090 10
(x1 − y1, x2 − y2 , x3 ) = R(sinφ cosθ, cosφ, sinφ sinθ), x3 → ∞ , (32) 

 Figure 4. Spherical coordinates 0 ≤θ ≤ π, 0 ≤φ ≤ π . 

where R = 

x − y . The wave number vector in expressed in a similar fashion as

(k1, k2 , k3 ) =κ o (sinφ′ cosθ′, cosφ′, sinφ′sinθ′) . (33) 

Using (32) and (33) in (31), and upon neglecting the flight effect, the phase factor becomes

Θ(



kt ,



x,ω) = Rκ o (sinφ sinφ′ cos(θ −θ′)+ cosφ cosφ′), M∞ = 0 , (34) 

where we have selected χ ∞ = −κ o sinφ′sinθ′ . The large parameter in applying the stationary 

phase technique [Ref. 19] is Rκ o 1 . This requires the source to observer distance R to be 

much larger than the wavelength of the radiated sound. If we define a Strouhal frequency as 

St =ωδ o /Ue , where δ o is the BL thickness (distance from the wall at which the mean velocity is 

0.99Ue), then the large parameter requirement implies St >> c∞δ o /(UeR) which is not a very 

stringent condition even at relatively low frequency.

The Jacobian in making coordinates transformation from ( k1, k2 ) to (θ′,φ′) is κ o

2 sinθ′sin2

φ′ or 

dk1dk2 =κ o

2 sinθ′sin2

φ′dθ′dφ′ . 

Using the new coordinates, the point of stationary phase (θ*

,φ*) is obtained when we set equal 

to zero the two partial derivatives of phase factor Θ. 

∂Θ

∂θ′

= 0, ∂Θ

∂φ′

= 0 . (35) 

which results in 

θ* = θ, φ* = φ . (36) 

NASA/CR—2013-218090 11
Equation (36) simply implies that in the absence of flight effect, the major contribution to the 

radiated sound field is due to wave numbers with magnitude κ o in the same direction as the 

observer point(θ,φ). Subsequently the contribution to the integral in (30) in the vicinity of the 

stationary point is obtained when we expand the phase Θ (to second order) about this point, and 

evaluate the rest of the integrand at the point of stationary phase 

kt

• = (k1•

, k2

•

). 



kt

• =κ o (sinφ cosθ, cosφ), 

G(



x,



y;ω)  i

(2π )

3

−ω

−ω + k1

•

U(y3 ( ))

2

b2 (



kt

•

,ω)V1(



kt

•

, y3,ω)

c∞c(y3 )Wo k1

• ( ,ω,Z ) ×

exp iκ oR 1− 1

2

(θ′ −θ)

2 sin2

φ − 1

2

(φ′ −φ) ⎧ 2

⎨

⎩

⎫

⎬

⎭

⎛

⎝

⎜ ⎞

⎠

⎟

φ′

∫

θ′

∫ (κ o

2 sinθ sin2

φ dθ′dφ′),

(37) 

as (R → ∞). The above double integral is now evaluated as

e

iκ oR

κ o

2 sinθ sin2

φ exp − iκ oR

2 (α2 sin2

φ + β2) ⎛

⎝

⎜ ⎞

⎠

⎟ −∞

∞

∫ −∞

∞

∫ dαdβ = −2πiκ o

sinθ sinφ

R

e

iκ oR . (38) 

Placing this result back in (37) shows 

G(



x,



y;ω)  − 1

(2π )

2

e

iκ oR

R

sinθ sinφ

c∞

2

c(y3 )

b2 (



kt

•

,ω)V1(



kt

•

, y3,ω)

Wo



kt

•

( ,ω,Z )

1

1− sinφ cosθ

U(y3 )

c∞

⎛

⎝

⎜

⎞

⎠

⎟

2 . (39) 

The GF G(



x,t;



y,τ ) is now obtained as 

G(



x,t;



y,τ ) = G(



x,



y;ω)

−∞

∞

∫ e−iω (t−τ )

dω . (40) 

Note that with k1 =κ o sinφ cosθ , the upper limit placed on k1 in section 2 will be satisfied at all 

subsonic conditions Uj / c∞ <1. At supersonic conditions only certain observer locations will be 

subject to the critical layer singularity.

4.0 Far-Field Acoustics 

 The far-field sound spectral density is evaluated as a FT of the pressure auto-covariance 

p

2

(



x,ω) = (γ p)

2

e

iωτ 1

2T

π ′

−T

T

∫ (



x,t)π ′(



x,t +τ )dt

⎡

⎣

⎢ ⎤

⎦

⎥dτ

−∞

∞

∫ . (41) 

where T denotes some large time. Upon placing (5) and (40) into (41), it is shown that [Ref. 20] 

NASA/CR—2013-218090 12
p

2

(



x,ω) = G∗(



x,



y−



ξ /2,ω

τ =−∞

∞

∫ 

ξ

∫ 

y

∫ )G(



x,



y+



ξ /2,ω)q(



y,



ξ,τ )e

iωt

dτd



ξd



y , (42) 

where q denotes a two-point space-time correlation between noise generating sources at points A 

and B separated by space 

ξ and timeτ . Using the source term in equation (1) we have 

q(



y,



ξ,τ ) = (γ p)

2 lim

T→∞

1

2T

Γ*(

−T

T

∫ 

y−



ξ/2,t)Γ(



y+



ξ/2,t +τ )dt . (43) 

Since the exponential phase factor in the GF equation (39) depends on the distance R =|



x−y |

between observer point 

x and source point 

y , the product of the GF and its conjugate (denoted 

by *) is evaluated at the center of the correlation multiplied by a phase factor exp(−i



k.



ξ ), where 

wave number 

k is directed as 

x−y and has a magnitude of κ o 

p

2

(



x,ω) = |G(



x,



y;ω)|

2



y

∫ q(



y,



ξ,τ )

τ =−∞

∞

∫ 

ξ

∫ e

iωτ −i



k .



ξ dτd



ξd



y . (44) 

 

As is a common practice in acoustic analogy noise-prediction models, the space-time FT of the 

source correlation function, denoted as the inner double integral in (44), is evaluated in closedform once an acceptable physics-based source model is obtained. The modeling becomes much 

less cumbersome if the spatial derivatives present in the source were to be transferred to the GF 

within equation (5) and prior to forming the sound spectral density function. Subsequently, the 

final GF as defined in (39) could be subject to new spatial derivatives at the source point 

y while 

source Γ(



y,t) would be free of similar derivatives. 

Since the GF depends on both V1 and V2, which in turn depend on ω ,θ , and φ , the pair (V1,V2 ) 

is determined numerically per pre-selected observer angles and frequency using the local BL 

mean velocity and density profiles. The integral over the source region in equation (44) is then 

evaluated to obtain the far-field noise as a contribution from independent correlation volume 

elements that comprise the source. As pointed out earlier, the volume integration is limited to 

regions with relatively stronger source strength while the mean flow is considered as locally 

parallel in both x1 and x2 directions. 

In the limit of very low Mach number ( M < 0.10 ), it is argued [Ref. 9] that there is a direct 

correspondence between the acoustic domain of the wall-pressure spectrum and the far-field 

spectral density of the radiated sound. This approximation ignores mean flow refraction and 

sound produced by the turbulence outside the boundary layer at distances that are not small 

relative to the acoustic wavelength. Following this argument, at large distance R from a hard 

wall region of area A, Howe [Ref. 21] proposed an expression for the sound spectral density in 

terms of the so-called blocked pressure, which in the present notation is 

p

2

(



x,ω) = Aκ o

2 sin2

θ sin2φ

R2 P(



kt ,ω) , (45) 

NASA/CR—2013-218090 13
where P(



kt ,ω) is the wall-pressure wave number-frequency spectrum, and is defined as a FT of 

the space-time pressure correlation function q(ξ1,ξ 2,τ )of the wall-pressure 

P(



kt ,ω) = 1

(2π )

3 q(ξ1,ξ 2 ,τ )exp iωτ − i



kt .



ξ ( t)dτ

A

∫ τ =−∞

∞

∫ dξ1dξ 2 . (46) 

Here 

ξt is the spatial separation vector of the correlation on the surface and 

kt is the 

corresponding wave-number vector. The wall-pressure correlation function over a rectangular 

area A is defined as

q(ξ1,ξ 2 ,τ ) = lim

T→∞

1

2T

1

A p′(y1, y2 ,0,t)

− L2

L2

∫ − L1

L1

∫ −T

T

∫ p′(y1 + ξ1 ,y2 + ξ 2 ,0,t +τ )dy1dy2dt . (47) 

Various models for P(



kt ,ω) have been proposed in the literature, however there is no universally 

accepted form for the wall-pressure spectrum that could directly be substituted into (45). For 

example low wave number measurements of Sevik [Ref. 10] suggest 

P(



kt ,ω)

ρ∞

2

v*

3

δ o

3 ≈ 127(Ue / c∞ )

2

(v* /Ue )

(ωδ o /Ue )

4.5 , 24 <

ωδ o

Ue

< 240, 0.01 <

Ue

c∞

< 0.15. (48) 

where v* ≈ 0.03Ue is the friction velocity, and δ o denotes the BL thickness as defined earlier. 

More complicated expressions have been suggested for the wall-pressure wave numberfrequency spectrum P(



kt ,ω) with additional parameters that are meant to account for the 

attenuation and refraction of sound in the BL [Ref. 22] and/or surface roughness [Ref. 23] and 

dependence on the individual wave number components on the surface. The validity of these 

empirical models remains to be confirmed experimentally. A comparison of semi-empirical 

equation (45) with the analytical solution given in equation (44) shows that the latter should 

account for all of the above-mentioned effects when source information as well as the GF are 

available within and outside the BL. Ideally, a RANS solution to the wall-bounded flow would 

provide both the mean flow and turbulence information required in evaluating the GF and the 

source correlation in a turbulent boundary layer and the main jet. 

5.0 Two-Dimensional Approximation 

 When mean flow and turbulence are independent of the transverse direction x2 (unlike in 

section 3 where the rectangular flow was limited to a finite x2 domain), the analysis is carried out 

in two dimensions. Following the steps outlined earlier, the new GF is obtained from an 

equation similar to (30) 

G(



x,



y;ω) = i

(2π )

2

1

c(y3 )c(x3 )

−ω + k1U(x3 )

−ω + k1

U(y3 ( ))

2

b2V1(k1, y3,ω)

Wo (k1,ω,Z) k1

∫ e

iΘ(k1,



x,ω )

dk1 , (49) 

NASA/CR—2013-218090 14
where phase parameter is now defined as

Θ(k1,



x,ω) = k1(x1 − y1 )− χ ∞ x3 , χ ∞

2 =κ o2

−k1

2 . (50) 

It is readily shown that the point of stationary phase is k1

i =κ o cosθ , and 

G(



x,



y;ω)  − e

iπ /4

(2π )

3/2

e

iκ oR

κ oR

sinθ

c∞

2

c(y3 )

b2 (k1

i

,ω)V1(k1

i

, y3,ω)

Wo k1

i ( ,ω,Z )

1

(1− U(y3 )

c∞

cosθ)

2

 . (51) 

The source correlation now uses 

ξ = (ξ1,ξ 3 ) as a separation vector, and the far-field sound is 

evaluated using equation (42) with the volume element as d



y = dy1dy3 , and per unit flow 

thickness in x2 direction. Note that both 3D and 2D expressions for the GF could be written in a 

similar from if equations (39) and (51) were normalized with respect to their corresponding freespace GF. 

6.0 Numerical Results 

Sample Green’s function calculations are presented using analytical representations of the jet 

mean velocity and temperature profiles in the proximity of a flat surface. 

The mean velocity profile adopted here depends on the normal direction y3 only, and is modeled 

for a jet with thickness Dj = 2 in 

U(η)

Uj

=

tanh(

Dj

η

d1

), η <1.05

1

2

+

1

2

tanh 1

d2

1/ 2

η −1

− η −1

1/ 2

⎛

⎝

⎜ ⎞

⎠

⎟ , η ≥1.05

 (52) 

Using parameters (d1, d2 ) = (0.10, 2.0) , the velocity profile, divided by jet velocity Uj , is plotted 

(Fig. 5) as a function of the non-dimensional distance from the wallη = y3 / Dj . The boundary 

layer thickness for this flow is δ o / Dj = 1.324d1 . 

NASA/CR—2013-218090 15
The mean static temperature is modeled as a composite of the two profiles – the first is obtained 

by placing the above velocity profile in Crocco-Busemann law 

T1(η)

T∞

= 1+ (TR −1)

U(η)

Uj

− γ −1

2

U(η)

c∞

⎛

⎝

⎜

⎞

⎠

⎟

2

,

 

(53a) 

where TR is the jet plenum stagnation temperature ratio. The second profile is intended to 

simulate frictional heat generation in the proximity of the wall

T2 (η)

T∞

= 1

d3

1

2

+

1

2

tanh 1

d4

( 1

Dj

η

− Djη) ⎛

⎝

⎜

⎞

⎠

⎟ . (53b) 

Using parameters (d3, d4 ) = (4.0, 3.0) the temperature profile T (η) = T1(η)+ T2 (η) is shown in 

Fig. 6 subject to jet exit values of Uj / c∞ = 0.90 , and TR = 3.0 . The ideal gas law c

2 = γ RT was 

used to construct the speed of sound as shown in Fig. 6. 

 Figure 5. Mean velocity profile.

NASA/CR—2013-218090 16
The GF is solved numerically at specific values of the Strouhal frequency Sto defined as 

Sto ≡ 1

2π

ωDj

Uj

 , (54) 

We define non-dimensional parameters κ =κ oDj = 2π Sto (Uj / c∞ ) , c = c / c∞ , U = U / c∞ , 

R = R / Dj , and divide the 3D and 2D expressions in (39) and (51) by their respective free-space 

values of(−e

iκ R / 4π R) and (−e

iκ R+iπ /4 / 8πκ R ). Additionally constant 1/(πc∞

3 ) is factored out 

in order to define a normalized GF applicable to the far field 

GN (η,ω) =

sinθ sinφ

c

b2 (



kt

*

,ω)V1(



kt

*

,η,ω)

Wo (



kt

*

,ω,Z)

1

(1−U sinφ cosθ )

2 , 3D

sinθ

c

b2 (k1

*

,ω)V1(k1

*

,η,ω)

Wo (k1

*

,ω,Z)

1

(1−U cosθ )

2

Dj

to

. 2D

 (55) 

Here Wo = Dj

Wo is dimensionless, and derivatives present in the Wronskian are now evaluated 

with respect to the normalized distance η . Factor (Dj / to ) present in the 2D solution implies 

that the pressure field is evaluated per unit flow thickness to in x2 direction. Since the 

Wronskian in equation (23a) is independent of normal distance to the wall, computationally it is 

advantageous if it were evaluated at infinity where V2 may be eliminated from the solution

Figure 6. Mean profiles for static temperature (solid line) 

 and sound speed (dashed line) at .

NASA/CR—2013-218090 17
b2 (



kt

*

,ω)

Wo (



kt

*

,ω,Z)

= − e

iDjχ ∞η

iDjχ ∞V1(



kt

*

,η,ω)+V′

1(



kt

*

,η,ω)

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥η→∞

. (56) 

Sample Computations

We start with a rigid boundary condition on the wall (Z → ∞) and a set of values for the 

observer location and flow condition as (φ = π /2,θ = π /4, Sto= 0.30,Uj /c∞= 0.90, TR = 3.0) . 

Two linearly independent solutions V1 and V2 to equation (23a) are shown in Figures 7 and 8. 

The first solution V1 is real, and possesses a relatively larger amplitude. The second solution V2

is complex, and is solved subject to specified boundary conditions at η = 0 and ∞ . The missing 

initial boundary condition is solved as V′

2 (0) = (7.15751, 0.06114i). At large distance η from 

the boundary, both V1 and V2 exhibit an oscillatory behavior with a regular period equal to 

2π /(−Djχ ∞ ). 

Constant b2 is determined from Eq. (29) as η → ∞. Figure 9 shows that 

b2 = (−0.18368, − 0.13127i) . The real and imaginary components of GN (η,ω) are presented in 

Fig. 10. Since the far-field pressure is evaluated according to equation (44) as the convolution of 

the source correlation function and the GF, the importance of the GF appears only where the 

source density is not insignificant. This, most likely, is limited to the initial several diameters 

normal to the wall where jet/wall turbulence and/or heat related sources are present. Figure 11 

displays an expanded view of the GF within the first 2 diameters normal to the wall. The 

amplitude is relatively larger at η = 1 compared to regions within the TBL, and the phase is 

constant at 0.623 Rad. 

 

Figure 7. Solution V1 (η) (solid line) and its derivative (dashed line) subject to the mean velocity and 

temperature profiles of Figs 5 and 6 at (φ = π / 2, θ = π / 4, Sto = 0.30, Uj / c∞ = 0.90,TR = 3.0).

1 2 3 6 h

-60

-30

30

60

V1HhL

V1'

V1

NASA/CR—2013-218090 18
Figure 8. Real (solid line) and imaginary (dashed line) components of solution V2 (η) subject to the

mean flow profiles of Figs 5 and 6 at (φ = π /2, θ = π /4, Sto=0.30, Uj /c∞ = 0.90,TR = 3.0).

Figure 9. Real (solid line) and imaginary (dashed line) components of constant 

 parameter b2 as η → ∞ . 

 

Figure 10. Real (solid line) and imaginary (dashed line) components of the Green’s function

 GN (η,ω) at (φ = π /2, θ = π /4, Sto=0.30, Uj /c∞ = 0.90,TR = 3.0).

1 2 3 6 h

-0.5

0.5

1

2

V2HhL

Im

Re

1 2 3 6 h

-1

1

2

V2HhL e‰ hDJ c•

Im

Re

1 2 3 6 h

-0.4

-0.2

0.2

0.4

GN

Im

Re

NASA/CR—2013-218090 19
 

Figure 11. Magnitude (solid line) and phase (dashed line) of the Green’s function GN (η,ω)

 at (φ = π /2, θ = π /4, Sto=0.30, Uj /c∞ = 0.90,TR = 3.0).

Next, we increase the frequency to Sto = 1.0 while the remaining parameters are kept the same as 

before. Noticeable reduction in the magnitude of the GF (see Fig. 12) indicates that sound 

generation within the TBL should be minimal relative to the previous example at Sto= 0.30

provided that the source densities remain comparable. 

 

In the third example, temperature effect is examined by considering an unheated jet at TR = 1.0 , 

while the remaining parameters are kept the same as in Fig. 11. The temperature profile for this 

flow is seen in Fig. 13, and the corresponding GF solution is presented in Fig. 14. The 

magnitude of the GF at and near the wall (η <1.1 ) is increased considerably (nearly four times) 

relative to the heated jet shown in Fig. 11. It is noted that according to equation (55), that 

temperature effect enters the computations through factors (c∞ /c) as well as solution V1(η). 

0.1 0.6 1.1 2 h

0.3

0.6

GN

Arg

Abs

Figure 12. Magnitude (solid line) and phase (dashed line) of the GF at 

 – with the remaining parameters the same as in Fig. 11.

NASA/CR—2013-218090 20
The effect of the observer location on the GF is examined by considering a polar angle of 

θ = π / 2 while the remaining parameters are kept the same as in example 1. Figure 15 shows an 

increase in the amplitude of the GF in a direction normal to the plate. In particular, there is 

substantial increase in the magnitude of the GF within the BL at η < 0.13 . 

Figure 13. Mean flow profiles for temperature

(solid line) and sound speed (dashed line) in the 

unheated jet at .

Figure 14. Magnitude (solid line) and phase 

(dashed line) of the Green’s function at

 Figure 15. Magnitude of the Green’s function at 

 (solid line), and (dashed 

 line), at conditions of Fig. 11.

NASA/CR—2013-218090 21
The effect of wall conditions may be examined by selecting numerical values for the normalized 

impedance function Z . The real and imaginary components of the surface impedance depend 

not only on the wall conditions, but also on the external flow characteristics such as Mach 

number and sound frequency and amplitude [Refs. 24, 25]. Aside from the solid boundary 

conditions (Z → ∞) examined earlier, here we choose a pair of values for the specific resistance 

and reactance as Z = (1± 0.50i). Computational results are shown in figure 16 at unheated 

conditionsTR =1.0 , and at a polar angle of θ =π /4 . The amplitude of the GF is affected slightly, 

and only within the first diameter normal to the wall. The phase (shown in radians) remains 

constant for the rigid wall, but becomes a function of the source location when the impedance 

function consists of a reactive component.

7.0 Summary 

 Prediction of the aircraft noise due to the interaction of the jet exhaust with a nearby solid 

surface is of increasing interest in the design of future civil transport. Concepts such as hybrid 

wing body aircraft, over the wing engine mount, or distributed propulsion are likely to contribute 

to jet surface interaction noise. While nearby surfaces could provide significant noise shielding 

Figure 16. The effect of wall impedance on the magnitude

(top) and phase (bottom) of the GF – (rigid: solid line);

(1 +0.50 i : dashed line); and (1 – 0.50 i : dash-dot) at

NASA/CR—2013-218090 22
in certain directions, they also increase noise in other directions. In this study, we presented a 

formal solution to the propagation Green’s function applicable to a rectangular jet exhaust in the 

proximity of a flat surface. The governing equation is the compressible Pridmore-Brown 

equation. Both frictional heat generation and thermal gradients in the boundary layer are 

potential contributors to noise generation and propagation. The general expression for the farfield noise (eq. 42) resembles that in the absence of a nearby surface – the major difference is 

concealed in the propagator, which should now be evaluated subject to prescribed boundary 

conditions on the surface. It was pointed out that aside from the scrubbing noise, scattered noise 

(or trailing edge noise) is also present on both side of the surface. A comprehensive prediction 

model needs to account for both noise components. 

Numerical results were presented that show a parametric study of the propagation Green’s 

function at selective values of the source frequency, jet temperature, observer angle, and wall 

impedance function using analytical representations of the mean flow profiles. For round jets, a 

conformal mapping to a rectangular strip needs to be carried out. In practice, a ReynoldsAveraged Navier-Stokes solution to the nozzle flow in the proximity of the surface (with an 

appropriate turbulence model) provides the mean flow and turbulence information required for 

modeling both momentum-flux and enthalpy-flux source components. The propagation equation 

should be solved numerically at each pair of observer angles (θ, φ), and at each frequencyω

subject to the local mean velocity and density profiles. In a 2D formulation, the GF is evaluated 

according to equation (51) while the mean flow is considered as a superposition of parallel slices 

in x1 direction and infinitely long in x2 direction. When the locally parallel flow approximation 

is extended to x2 direction the GF is evaluated according to equation (39). Here, the jet may be 

sub-divided into elements in the span-wise x2 direction in order to account for a finite wetted 

domain with an eventual decay of the source density in this direction. 

Nomenclature 

b2 Far field amplitude (eq. 29)

c Sound speed

Dj Jet Diameter

δ o Boundary layer thickness

η Normalized distance y3 / Dj

G Green’s function

Gˆ,



G Transformed Green’s function

GN Normalized Green’s function

h′ Enthalpy fluctuations

h Enthalpy

h′

o moving frame stagnation enthalpy ( h′ + v′

j v′j / 2 )

NASA/CR—2013-218090 23
η Normalized distance from the wall 

γ Specific heats ratio

Γ Source 

k Wave number 

kt Wave number vector (k1, k2 )

κ o Wave number magnitude (ω/c∞ )

M Mach number (U/ c)

ω Radian frequency (2π f )

π′ Normalized pressure fluctuation

p′ Acoustic pressure

P(kt ,ω) Wall-pressure wave number-frequency spectrum

q(



y,



ξ,τ ) Source correlation function

ρ Density

R Distance |



x −y| 

ξ Spatial separation vector

ψ Wall function (eq. 21b)

t Time

TR Stagnation temperature ratio

τ Time delay

U Mean axial velocity

Ue Peak velocity in the boundary layer

v′

i Turbulent velocity component

vi Mass-averaged velocity component

θ Polar angle 

φ Azimuthal angle

Wo Wronskian 

x Rectangular coordinates 

y Source location

Z Surface impedance

Subscripts

o At the surface variable

∞ Ambient condition

Superscripts

– Time average

^ Fourier transformed variable

~ Favre average (q = ρq / ρ)

NASA/CR—2013-218090 24
Appendix-A Influence of the Solid Surfaces 

The Navier-Stokes equations when linearized about a non-radiating base flow may be written as 

a set of five linear equations [Ref. 15] 

Lνμuμ = f

ν (



x,t), (μ,ν = 1,2,...,5) (A1) 

where uμ denotes the five dependent variables

uμ = (mi, p′

e, ρ′)

mi = ρv′

i , i =1, 2,3 (A2) 

(Latin indices repeat from 1 to 3, and Greek indices repeat from 1 to 5). The first three equations 

in (A1) correspond to the linearized momentum equation, followed by the energy and mass 

equations in the fourth and fifth place. The Green’s Function (GF) to set of equations (A1) with 

a non-zero delta function source placed in the σth equation is denoted as gμσ 

Lνμgμσ (



x,t |



y,τ ) = δ νσδ (



x − y)δ (t −τ ). (A3) 

The source term appearing on the right hand side of (A1) is

f

ν (



x,t) = ∂

∂xj

e′′ν j +δ ν 4 (γ −1)e′′

ij

∂vi

∂xj

, (i, j = 1,2,3), ν = 1,...,5 (A4) 

e′′ν j = −(ρv′

νv′j − ρv′νv′j),

v′

4 = (γ −1)(h′ +

1

2

v′

2

), v′

5 = 0.

(A5) 

The first three source components relate to difference between momentum flux terms and their 

Favre-averaged values, while the fourth term is related to the enthalpy flux and its Favreaveraged value, and the fifth source term (mass equation) is obviously zero. 

Note that both momentum variable mi and pressure variable p′

e are non-linear, however, in 

practice we set mi  ρv′

i , and 

p′

e = p′ − (γ −1)

2

e′′

kk , (A6) 

implies that p′

e  p′ in the acoustic domain where turbulence is zero.

The field variables of interest are obtained from equations (A1) and (A3) using the source/GF 

convolution integral 

NASA/CR—2013-218090 25
uμ(



x,t) = gμν 

y

∫

τ

∫ (



x,t |



y,τ )f

ν (



y,τ )d



ydτ , (A7) 

For example, when μ = 4 the far-field acoustic pressure is evaluated as

p′(



x,t) ≈ p′

e (



x,t) = g4ν 

y

∫

τ

∫ (



x,t |



y,τ )f

ν (



y,τ )d



ydτ . (A8) 

Considering that f5 = 0 , the latter operation requires solving (A3) four times, each time by 

placing a delta function in the one of the first four equations. This process would also solve for 

other GF components that are not necessary for the purpose of evaluating p′ . It is 

computationally advantageous if the field variables were evaluated from the set of the adjoint 

equations that are governed by 

Lνμ

(a) gμσ(a)

(



x,t |



y,τ ) = δ νσδ (



x − y)δ (t −τ ). (A9) 

The adjoint operator (denoted by superscript a) is readily evaluated once we multiply equation 

(A1) by gνσ

(a)

(



x,t |



y,τ ), and rearrange its left hand side to appear as

gνσ

(a)

(



x,t |



y,τ )Lνμuμ(



x,t) = uμ(



x,t)Lμν

(a)

gνσ

(a)

(



x,t |



y,τ )+

∂

∂t

uμgμσ

(a)

( ) +

∂

∂xj

Fjσ . (A10) 

The last two terms on the RHS of (A10) represent the bi-linear form of uμ and gμσ

(a)

, and 

Fjσ (



x,t |



y,τ ) = vj(



x,t)uμ(



x,t)gμσ

(a)

(



x,t |



y,τ )+ u4 gjσ

(a)

+ uj g5σ

(a)

+ c

2

g4σ

(a) ( ). (A11) 

We solve for uσ (



y,τ ) by placing (A9) into (A10), and integrating this equation within a large 

four-dimensional space-time volume (



x,t)surrounding the source 

gνσ

(a)



x

∫ (



x,t |



y,τ )f

ν (



x,t)d



x dt

t

∫ = uσ (



y,τ )+

∂

∂t

uμgμσ

(a)

( ) +

∂

∂xj

Fjσ

⎛

⎝

⎜

⎞

⎠

⎟ 

x

∫

t

∫ d



xdt . (A12) 

Following the divergence theorem, the volume integral on the RHS of (A12) may be converted 

into a surface integral. When there are no solid surfaces to interfere with the sound, this integral 

is evaluated on the surface of a large 4D sphere in (



x,t) domain – resulting in a null contribution 

due to the vanishing of the field variables uμ on such a surface. Consequently (A12) becomes 

uμ(



x,t) = g(a)

νμ 

y

∫

τ

∫ (



y,τ |



x,t)f

ν (



y,τ )d



ydτ . (No solid surfaces) (A13) 

Equations (A7) and (A13) point to the reciprocity of the GF in the absence of solid surfaces, i.e., 

NASA/CR—2013-218090 26
gνμ

(a)

(



y,τ |



x,t) = gμν (



x,t |



y,τ ). (A14) 

By setting μ = 4 we evaluate the far-field pressure. It is seen that the advantage of (A13) over 

(A7) is that we now place a source only in the fourth equation within set (A9) to compute the 

required GF components for pressure. Equation (A13) represents the fundamental solution to the 

governing equations. In the presence of solid boundaries in a finite domain, the general solution, 

when formulated in terms of the adjoint GF, is complemented with surface integrals as shown in 

(A12). The reciprocity condition (A14) would not necessarily be satisfied when reflecting 

boundaries are present in the acoustic medium unless surface integrals are forced to disappear by 

imposing conditions on the adjoint GF. 

To investigate the surface integrals further, we allow volume 

x(t), externally and/or internally, 

be bounded by surface s(t)with a unit outward normal 

n (Fig. A1). The first term within the 

bracket on the RHS of (A12) is expressed as the sum of two terms following the Leibniz’s rule 

∂

 ∂t x(t)

∫ uμgμσ

(a)

(



x,t |

 ( y,τ ))d

x = d

dt

uμgμσ

(a)

(



x,t |



y,τ ) 

x(t)

∫ d



x − n ⋅



Vs

s(t)

∫ uμgμσ

(a)

(



x,t |



y,τ )ds . (A15) 

Figure A1. Solid surfaces within the acoustic medium

where 

V s

(



x,t)is the velocity at an arbitrary point on the surface s(t). We now integrate both 

sides of (A15) within the time interval [-T, T] for some large time T. The first terms on the RHS 

of (A15) vanishes due to the initial condition (in a remote past time), and the causality condition. 

Inserting (A15) into (A12) and applying the divergence theorem to the last term in (A12), we 

find (upon interchanging dummy variables) 

uμ(



x,t) = gνμ

(a)



y

∫

τ

∫ (



y,τ |



x,t)f

ν (



y,τ )d



ydτ + dτ

−T

T

∫ nj

s(τ )

∫ Vj

s

uνgνμ

(a)

(s,τ |



x,t) − Fjμ(s,τ |

 ⎡ x,t) ⎣ ⎤

⎦ds (A16) 

Integral equation (A16) represents a formal solution to the problem. The two integrals represent 

contributions to the acoustic field due to the volume and surface sources, respectively. In 

particular, the acoustic pressure is solved when μ = 4 . Moving the partial derivatives from 

NASA/CR—2013-218090 27
source to the GF and/or imposing boundary conditions on the surface integrals may achieve 

further simplifications. For example, placing f

ν from (A4) into (A16), and using the identity 

gνμ

(a)

(



y,τ |



x,t) ∂

∂yj

e′′ν j = ∂

∂yj

gνμ

(a)

(



y,τ |



x,t)e′′ ( ν j) − e′′ν j

∂

∂yj

gνμ

(a)

(



y,τ |



x,t)

followed by application of the divergence theorem on the first term on the right shows that

uμ(



x,t) = − ∂gνμ

(a)

∂yj

− (γ −1)

∂v

ν

∂yj

g4μ

(a) ⎛

⎝

⎜

⎞

⎠

⎟ 

y

∫

τ

∫ e′′ν j(



y,τ )d



ydτ + dτ

−T

T

∫ nj

s

∫ gνμ

(a)

(s,τ |



x,t)e′′ν jdsdτ

+ dτ

−T

T

∫ nj

s

∫ Vj

s

uνgνμ

(a)

(s,τ |



x,t) − Fjμ(s,τ |

 ⎡ x,t) ⎣ ⎤

⎦ds.

(A17) 

Further, substituting for Fjμ from (A11) into this last expression and imposing the no-slip 

boundary condition on a hard surface 

(Vj

s − vj).nj = 0 , (A18) 

results in the following integral equation for pressure 

p′

e (



x,t) = − Γν j

(a)



y

∫

τ

∫ (



y,τ |



x,t)e′′ν j(



y,t)d



ydτ + dτ

−T

T

∫ nj

s

∫ gν 4

(a)

(s,τ |



x,t)e′′ν jds

− dτ

−T

T

∫ nj

s

∫ p′

e (s,τ )gj 4

(a)

(s,τ |



x,t)+ mj(s,τ )(g54

(a)

+ c

2

g44

(a) ⎡ ) ⎣ ⎤

⎦ds

(A19) 

where 

Γν j

(a) ≡ ∂gν 4

(a)

∂yj

− (γ −1)

∂v

ν

∂yj

g44

(a)

. (A20) 

NASA/CR—2013-218090 28
Appendix-B Vanishing of the Surface Integrals 

Here we show that the boundary condition (21) eliminates the surface integrals in the Green’s 

function formulation of the acoustic field for bounded media. We start with equation (1), and 

apply Fourier transform with respect to variables x1, x2 and t as defined in eq. (7) with the over 

hat carrot denoting a transformed variable 

∂2

θˆ

∂x3

2 + f (



kt , x3,ω)θˆ = Λˆ , (B1) 

where function f was defined in eq. (11) and 

θˆ(



k, x3,ω) ≡ πˆ′(



kt , x3,ω) c(x3 )

−ω + k1

U(x3 )

, Λˆ (



k, x3,ω) ≡ iΓˆ

c(x3 ) −ω + k1

U(x3 ( ))

2 . (B2) 

Multiply eq. (9) by θˆ , and equation (B1) by 

G , subtract the two expressions, and integrate the 

result with respect to x3 



G ∂2

θˆ

∂x3

2 −θˆ ∂2 

G

∂x3

2

⎛

⎝

⎜

⎞

⎠

⎟

0

∞

∫ dx3 = 

GΛˆ −θˆ δ (x3 − y3 ( ))

0

∞

∫ dx3, (B3) 

or 

− G ∂θˆ

∂x3

−θˆ ∂



G

∂x3

⎛

⎝

⎜

⎞

⎠

⎟

0

∞

= θˆ(



k, y3,ω)− G(



kt

0

∞

∫ , x3, y3,ω)Λˆ (



k, x3,ω)dx3 . (B4) 

Upon recognizing that ∂θˆ / ∂x3 = −iχ ∞θˆ, and ∂



G / ∂x3 = −iχ ∞



G at infinity, the upper limit of the 

expression on the left hand side of (B4) vanishes. On the right, since the GF is self-adjoint we 

switch x3 and y3 in the argument of 

G , and subsequently we substitute expressions for Λˆ from 

(B2), and 

G from (10) into (B4) to show that in view of (19), the right hand side of this equation 

also vanishes. 

Consequently we arrive at 



G ∂θˆ

∂x3

−θˆ ∂



G

∂x3

= 0, x3 = 0 . (B5) 

Using surface boundary condition (18) in (B5) shows that 

∂



G

∂x3

−ψ 

G = 0, x3 = 0 (B6) 

where function ψ (k1,ω,Z) is given in (21b). 

NASA/CR—2013-218090 29
References 

[1] Brown, C. A., “Jet-Surface interaction test: far-field noise results,” ASME paper GT2012-69639, June 2012. [2] Podboy, G. G., “Jet-Surface interaction test: phased array noise source localization results,” ASME paper GT2012-69801, 

June 2012. [3] FFowcs Williams, J. E., and Hall, L. H., “Aerodynamic sound generation by turbulent flow in the vicinity of a scattering half 

plane,” J. Fluid Mechanics, 40(4), (1970), pp. 657–670.

[4] Powell, Alan, “On the aerodynamic noise of a rigid flat plate moving at zero incidence,” J. Acoustical Soc. Am., 31(12), 

(1959), pp. 1649-1653.

[5] Lawrence, J. L. T., Azarpeyvand, M., and Self, R. H., “Interaction between a flat plate and a circular subsonic jet, “ AIAA 

paper AIAA-2011-2745, June 2011.

[6] Bowman, J. J., Senior, T. B. A., and Uslenghi, P. L. E., Electromagnetic and Acoustic Scattering by Simple Shapes (1987),

Hemisphere Publishing Corp., New York.

[7] Howe, M. S., “A review of the theory of trailing edge noise,” J. Sound and Vib. 61(3), (1978), pp. 437-465.

[8] Goldstein, M. E., “Scattering and distortion of the unsteady motion on transversely sheared mean flows,” J. Fluid Mechanics, 

91(4), (1979), pp. 601-632.

[9] Howe, M. S., “Surface pressure and sound produced by turbulent flow over smooth and rough walls,” J. Acoustical Soc. Am., 

90(2), (1991), pp. 1041-1047.

[10] Sevik, M. M., “Topics in hydrodynamics,” Proceedings of IUTAM Symposium, Aero- and Hydrodynamics, Lyon (1985), 

Berlin: Springer-Verlag.

[11] Pridmore-Brown, D. C., “Sound propagation in a fluid flowing through an attenuating duct,” J. Fluid Mech. 4, (1958), pp. 

393–406.

[12] Goldstein, M. E., “Relation between the generalized acoustic analogy and Lilley’s contribution to aeroacoustics,” 

International J. Aeroacoustics, 9(4, 5), (2010), pp. 401-418.

[13] Chase, D. M., and Noiseux, C. F., “Turbulent wall pressure at low wavenumbers: Relation to nonlinear source in planar and 

cylindrical flow,” J. Acoust. Soc. Am., 72(3), (1982), pp. 975–982.

[14] Bark, F. H., “On the wave structure of the wall region of a turbulent boundary layer,” J. Fluid Mechanics, 70, (1975), pp. 

229–250.

[15] Goldstein, M. E., “A generalized acoustic analogy,” J. Fluid Mechanics, 488, (2003), pp. 313-333.

[16] Chandiramani, K. L. “Diffraction of evanescent waves, with application to aerodynamically scattered sound and radiation 

from unbaffled planes,” J. Acoustical Soc. Am. 55(1), (1974), pp. 19-29.

[17] Howe, M. S., “The displacement-thickness theory of trailing edge noise,” J. Sound and Vib. 75(2), (1981), pp. 239-250.

[18] Jones, D. S., “The scattering of sound by a simple shear layer,” Proc. Roy Soc. A, 284, (1977), pp. 287–328.

[19] Erdelyi, A., Asymptotic Expansions (2010), Dover Publications, Inc.

[20] Khavaran, A., Bridges, J., and Georgiadis, N., “Prediction of turbulence-generated noise in unheated jets,” NASA/TM–

2005-213827, 2005.

[21] Howe, M. S., “On the generation of sound by turbulent boundary layer flow over a rough wall,” Proc. R. Soc. London, A 

395, (1984), pp. 247-163

[22] FFowcs Williams, J. E., “Boundary layer pressure and the Corcos model: A development to incorporate low wavenumber 

constraints,” J. Fluid Mechanics, 125, (1982), pp. 9-25.

[23] Chase, D. M., “The character of turbulent wall pressure spectrum at sub-convective wavenumbers and a suggested 

comprehensive model,” J. Sound and Vib. (112), (1987), pp.125-147.

[24] Watson, W. R., Tracy, M. B., Jones, M. G., and Parrott, T. L., “Impedance eduction in the presence of shear flow,” AIAA 

paper AIAA-2001-2263, May 2001.

[25] Malmary, C., Carbonne, S., Auregan, Y., and Pagneux, V., “Acoustic impedance measurement with grazing flow,” AIAA 

paper, AIAA-2001-2193, May 2001.

NASA/CR—2013-218090 30

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Jet Surface Interaction--Scrubbing Noise 

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

Generation of sound due to scrubbing of a jet flow past a nearby solid surface is investigated within the framework of the generalized 

acoustic analogy theory. The analysis applies to the boundary layer noise generated at and near a wall, and excludes the scattered noise 

component that is produced at the leading or the trailing edge. While compressibility effects are relatively unimportant at very low Mach 

numbers, frictional heat generation and thermal gradient normal to the surface could play important roles in generation and propagation of 

sound in high speed jets of practical interest. A general expression is given for the spectral density of the far-field sound as governed by the 

variable density Pridmore- Brown equation. The propagation Green’s function should be solved numerically starting with the boundary 

conditions on the surface and subject to specified mean velocity and temperature profiles between the surface and the observer. The 

equivalent sources of aerodynamic sound are associated with non-linear momentum flux and enthalpy flux terms that appear in the

linearized Navier-Stokes equations. These multi-pole sources should be modeled and evaluated with input from a Reynolds-Averaged 

Navier-Stokes (RANS) solver with an appropriate turbulence model. 

15. SUBJECT TERMS

Acoustics; Noise; Jet noise; Propulsion noise 

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